Applied Statistics and Probability for Engineers

630 CHAPTER 16 STATISTICAL QUALITY CONTROL

16-9 CONTROL CHART PERFORMANCE

Specifying the control limits is one of the critical decisions that must be made in designing a

control chart. By moving the control limits further from the center line, we decrease the risk

of a type I error—that is, the risk of a point falling beyond the control limits, indicating an

out-of-control condition when no assignable cause is present. However, widening the control

limits will also increase the risk of a type II error—that is, the risk of a point falling between

the control limits when the process is really out of control. If we move the control limits closer

to the center line, the opposite effect is obtained: The risk of type I error is increased, while the

risk of type II error is decreased.

The control limits on a Shewhart control chart are customarily located a distance of plus

or minus three standard deviations of the variable plotted on the chart from the center line.

That is, the constant kin equation 16-1 should be set equal to 3. These limits are called

3-sigma control limits.

A way to evaluate decisions regarding sample size and sampling frequency is through the

average run length (ARL)of the control chart. Essentially, the ARL is the average number of

points that must be plotted before a point indicates an out-of-control condition. For any Shewhart

control chart, the ARL can be calculated from the mean of a geometric random variable

(Montgomery 2001). Suppose that pis the probability that any point exceeds the control limits. Then

ARL (16-28)

1

p

Thus, for an chart with 3-sigma limits, p0.0027 is the probability that a single point falls

outside the limits when the process is in control, so

is the average run length of the chart when the process is in control. That is, even if the process

remains in control, an out-of-control signal will be generated every 370 points, on the average.

Consider the piston ring process discussed in Section 16-4.2, and suppose we are sampling

every hour. Thus, we will have a false alarmabout every 370 hours on the average. Suppose we

are using a sample size of n5 and that when the process goes out of control the mean shifts to

74.0135 millimeters. Then, the probability that falls between the control limits of Fig. 16-3 is

equal to

Therefore, pin Equation 16-28 is 0.50, and the out-of-control ARL is

ARL

1

p

1

0.5

 2

P 3  6 Z 04 0.5

P c

73.986574.0135

0.0045

Z

74.013574.0135

0.0045

d

P 3 73.9865X74.0135 when 74.0135 4

X

X

ARL

1

p

1

0.0027

 370

X

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